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First-order linear equations

Problem 6.125 · medium

Solve \( \displaystyle y' + 1y = 3 e^{- x} \) with \( \displaystyle y(0) = -1 \).

The equation is linear in standard form; the integrating factor is e^(∫1 dx) = e^(1x).
\[ \frac{d}{d x} Y{\left(x \right)} e^{x} = Y{\left(x \right)} e^{x} + e^{x} \frac{d}{d x} Y{\left(x \right)} \]
Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
\[ \int 3\, dx = 3 x \]
Integrate the right side.✓ Proved
Setting x = 0 and y = -1 fixes the constant of integration: C = -1.
\[ \left(3 x - 1\right) e^{- x} + \frac{d}{d x} \left(3 x - 1\right) e^{- x} = 3 e^{- x} \]
The solution satisfies the equation.✓ Proved
\[ -1 \]
And the initial condition.✓ Proved

Answer \( y = \left(3 x - 1\right) e^{- x} \)

Lines: 4 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Not checked	—	a sentence; read, not computed
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	sympy.checkodesol substitutes the solution back; y(0) matches

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution skips the crucial step of integrating the right-hand side (which is 3*e^x, not 3) and determining the constant of integration. It jumps from the derivative form to the final answer without showing the integration of 3*e^x to get 3x*e^x + C, nor does it correctly derive C=-1 from the intermediate step. The sentence in step 4 claims C=-1 is fixed by the initial condition, but the algebraic steps connecting the integrated form to the final answer are missing, making the derivation incomplete and logically disjointed.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution skips the crucial step of integrating the right-hand side (which is 3*e^x, not 3) and determining the constant of integration. It jumps from the derivative form to the final answer without showing the integration of 3*e^x to get 3x*e^x + C, nor does it correctly derive C=-1 from the intermediate step. The sentence in step 4 claims C=-1 is fixed by the initial condition, but the algebraic steps connecting the integrated form to the final answer are missing, making the derivation incomplete and logically disjointed.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The integrating factor is calculated as e^x, but the subsequent steps (integration and final solution) incorrectly use e^{-x} as the integrating factor or fail to divide by it properly. Specifically, step 3 integrates 3 instead of 3e^x, and step 4 derives C=-1 based on the incorrect form y = 3x + C rather than the correct y = (3x + C)e^{-x}. The stated answer is actually correct for the problem, but the derivation shown is mathematically inconsistent and wrong.
gpt-oss:20b: pass 2026-09-27

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/linear_first_order, checked 2026-09-27 with SymPy 1.14.0.